It is clear from the respective definitions that every solution to a
problem \(({\tt SVE})\)  is a fortiori a solution to the problems \(({\tt VE})\)  and \(({\tt WVE})\) ,
defined by the same data (whence the terminology). Of course, elementary examples
show that the converse is not true. If, in particular, \(\mathbb {Y}=\mathbb {R}\)  and
\(C=[0,+\infty )\) , then \(({\tt SVE})\) , \(({\tt VE})\)  and \(({\tt WVE})\)  collapse to the same problem,
namely what is called equilibrium problem after Blum and Oettli.
By their seminal paper [1]}, they definitely contribute
to popularize this kind of problem, in stressing its unifying feature and
undertaking a thorough study of it.
In fact, equilibrium problem revealed to be a convenient format
to treat in a unified framework various problems which are relevant
in operations research and mathematical programming, such as single and
multicriteria optimization problems, saddle point problems, complementarity
problems, variational inequalities, fixed point problems, Nash equilibrium
problems. In a similar manner, vector equilibrium problems
provide a format able to subsume vector optimization problems,
vector complementarity problems and vector variational inequalities
(see [2]}, [3]}, [4]} and references therein).
For this reason, in the last two decades vector equilibrium problems
became the subject of many investigations. As it is reasonable,
within the fast growing literature in this area, a remarkable
amount of research work focussed on solution existence and related issues
(see, for instance, [5]}, [6]}, [2]}, [3]}, [9]}, [4]}).
One of the main techniques of analysis in this context consists in
adapting to the vector case the approach due to Ky Fan, originally
proposed for scalar equilibrium problems (see, for instance,
[11]}). To achieve the nonemptiness
of the solution set, regarded as the intersection of a proper family
of sets, this approach leads to apply the Knaster-Kuratowski-Mazurkiewicz
theorem (a.k.a. Three Polish theorem) or some variant of it
(see [9]}, [13]}). Other approaches to solution existence
specific for strong vector equilibrium problems rely on different techniques,
such as the employment of separation theorems for convex sets (see [4]})
or the Kakutani fixed point theorem (see [6]}).
